{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 23 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE " Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Dark Red Emphasis" -1 259 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "" -1 264 "" 0 10 0 0 0 0 0 0 0 0 0 0 3 0 0 1 }{CSTYLE "Purple Emphasis" -1 265 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 266 "Tim es" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 272 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Ti mes" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal " -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 72 "A procedure for solving 2nd order linear DE's with constant coefficients" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 10.10.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "load " } {TEXT 0 7 "desolve" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 17 "The Maple m-file " }{TEXT 272 7 "DEsol.m" }{TEXT -1 32 " is required by t his worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 121 "It can be read into a Maple session by a command similar to the one that follows, where the file path gives its location." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "read \"K:\\\\Maple/procdrs/DEsol.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 74 "A procedure for solving 2nd or der linear DE's with constant coefficients: " }{TEXT 0 9 "desolveCC" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 16 "desolveCC: usage" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 260 18 "Calling Sequence:\n" }}{PARA 0 "" 0 "" {TEXT -1 17 " desolveC C( de )" }}{PARA 0 "" 0 "" {TEXT -1 52 " desolveCC( de ,y(x) )\n des olveCC( \{de,cnstrts \}) " }}{PARA 0 "" 0 "" {TEXT -1 34 " desolveCC( \{de,cnstrts \},y(x) ) " }{TEXT 262 1 "\n" }{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 11 "Parameters:" }}{PARA 0 "" 0 "" {TEXT -1 5 " \+ " }}{PARA 0 "" 0 "" {TEXT 23 9 " de - " }{TEXT -1 70 " a 2nd or der linear differential equation with constant coefficients," }}{PARA 0 "" 0 "" {TEXT -1 65 " that is, one which can be w ritten in the form" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+b;" "6#,& *(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(%\"bGF(" }{TEXT -1 1 " \+ " }{XPPEDIT 18 0 "dy/dx+c*y = f(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*& %\"cGF'%\"yGF'F'-%\"fG6#%\"xG" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 117 " with the first and 2nd order derivati ves entered as diff(y(x),x) and diff(y(x),x$2) respectively." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 4 "Note" } {TEXT -1 82 ": The dependent variable,say y, must be entered as y(x) e verywhere in the equation" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 23 12 "cnstrts - " }{TEXT 263 86 "t wo constraints involving the dependent variable, or its derivative, gi ven in the form" }}{PARA 0 "" 0 "" {TEXT 264 53 " \+ y(a)=b or D(y)(a)=b " }}{PARA 0 "" 0 "" {TEXT -1 67 " \+ so that \{de,cnstrts\} is a set of three equations." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 12 "Descrip tion:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " The procedure " }{TEXT 0 9 "desolveCC" }{TEXT -1 87 " attempts to solv e a 2nd order linear differential equation with constant coefficients. " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 261 8 "Opti ons:" }{TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 80 "info=true/false \nWith the option \"info=true\" the steps in the solution are shown." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 265 16 "How to activate:" }{TEXT 256 1 "\n" } {TEXT -1 155 "To make the procedure active, open the subsection, place the cursor anywhere after the prompt [ > and press [Enter].\nYou can then close up the subsection." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 26 "desolveCC: implementation " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28363 "desolveCC := proc()\n local ff ,de,ic0,ic1,x0,y0,x1,y1,derivs,df,df1,df2,df3,x,y,yx,\n x2,x3,yx2 ,pol,vars,d0,d1,d2,c0,c1,c2,tt,lsic,initcond,\n homog,h1x,h2x,tem p,prntflg,bs,fac,m1,m2,p,q,numcoeff,\n Options,j,bb,dsc,soln,eqrt s,e1,e2,e3,const1,const2,yy,\n eq0,eq1,cvals,t0,t1,realrts,uhx,hx ,cx,dcx,k1x,k2x,u1x,\n u2x,v1x,v2x,px,tx,dpx,u1OK,u2OK,sm,i,term, addterm,m,\n dscsq,cmplxrts,rootstype,startopts,xx,ee,signm,omit, p2x,\n ratfact,algfact,cd,Cs,gt,jS,j1,j2,s,nvars,order,a,la;\n \+ global C;\n\n# procedure to determine the sign of a real constant\nsig nm := proc(x)\n local i,s;\n s := signum(x);\n if type(s,integer ) then return s else do\n s := signum(evalf(x));\n if \+ type(s,integer) then return s end if;\n Digits := Digits + 5; \n if Digits>110 then break end if;\n end do;\n end if; \n return 'signm'(x);\nend proc: # of signm\n\n# extract common deno m of rational factors from alg expr or list\nratfact := proc(ff)\n l ocal fact,rfact,terms,rterms,i;\n if op(0,ff)=`list` then\n rte rms := NULL;\n for i to nops(ff) do\n rterms := rterms,ra tfact(ff[i])\n end do;\n return ilcm(rterms); \n elif op(0 ,ff)=`*` then\n fact := [op(ff)];\n rfact := NULL;\n fo r i to nops(fact) do\n if type(fact[i],rational) then \n \+ rfact := rfact,fact[i]\n end if;\n end do;\n i f nops([rfact])=1 then return denom(abs(rfact))\n else return 1 e nd if;\n elif op(0,ff)=`+` then\n terms := [op(ff)];\n rte rms := NULL;\n for i to nops(terms) do\n rterms := rterms ,ratfact(terms[i])\n end do;\n return ilcm(rterms);\n elif type(ff,rational) then\n return denom(abs(ff))\n else \n \+ return 1\n end if;\nend proc; # of ratfact\n\n# extract algebraic co mmon denom from alg expr or list\nalgfact := proc(ff,x)\n local i,fa ct,build,C,ee,t;\n if op(0,ff)=`list` then\n C := table();\n \+ ee := add(C[i]*ff[i],i=1..nops(ff));\n else\n ee := ff;\n \+ end if; \n fact := factor(ee);\n build := 1; \n if op(0,fact )=`*` then\n for i to nops(fact) do\n t := op(i,fact);\n \+ if numer(t)=1 and not has(denom(t),x) then\n build \+ := build*denom(t);\n end if;\n end do;\n end if;\n bu ild;\nend proc;\n\n if nargs>0 then \n ff := args[1]\n else\n error \"at least one argument must be supplied\"\n end if;\n \+ initcond := false;\n if type(ff,\{set(equation),list(equation)\}) a nd nops(ff)=3 then\n ff := map(_u -> if has(_u,D@@2) then convert (_u,diff) else _u end if,ff);\n de := op(1,ff);\n ic0 := op( 2,ff);\n ic1 := op(3,ff);\n if not has(de,diff) then\n \+ de := op(2,ff);\n ic0 := op(1,ff);\n end if;\n if not has(de,diff) then\n de := op(3,ff);\n ic1 := op(2 ,ff);\n end if;\n initcond := true;\n elif type(ff,equatio n) then\n de := ff;\n else\n error \"the 1st argument, %1, is invalid .. it should be an equation or a set (or list) of 3 equati ons\",ff;\n end if;\n\n startopts := 2;\n if nargs>1 then\n \+ ee := args[2];\n if type(ee,function) and nops(ee)=1 then\n \+ yy := op(0,ee);\n xx := op(1,ee);\n if type(xx,nam e) and type(yy,name) then\n startopts := 3;\n else \n error \"the 2nd argument, %1, has incorrect form for the dependent variable\",ee;\n end if;\n end if;\n end if; \n\n prntflg := false;\n if nargs>=startopts then\n Options:= [args[startopts..nargs]];\n if not type(Options,list(equation)) t hen\n error \"each optional argument must be an equation\"\n \+ end if;\n if hasoption(Options,'info','prntflg','Options') th en \n if prntflg<>true then prntflg := false end if;\n en d if;\n if nops(Options)>0 then\n error \"%1 is not a val id option for %2\",op(1,Options),procname;\n end if;\n end if; \n \n # Check out the derivatives in the DE.\n derivs := indets( de,'specfunc(anything,diff)');\n if derivs=\{\} then\n error \" the 1st argument, %1, is invalid .. it should be a differential equati on or a set (or list) containing a differential equation and two initi al conditions\",ff;\n end if;\n nvars := nops(indets(derivs,name)) ;\n if nvars<>1 then\n if nvars=0 then\n error \"there \+ is a problem with the independent variable occurring in the derivative (s)\";\n else\n error \"there should only be one independ ent variable in the differential equation\"\n end if;\n end if; \n nvars := nops(indets(derivs,anyfunc(name)));\n if nvars<>1 then \n if nvars=0 then\n error \"there is a problem with the \+ dependent variable occurring in the derivative(s)\"\n else\n \+ error \"there should only be one dependent variable in the differe ntial equation\"\n end if;\n end if;\n\n order := nops(derivs );\n if order=1 then\n error \"the differential equation shoul d have order 2\" \n elif order>2 then\n error \"there are too m any derivatives in the differential equation .. note that the differen tial equation should have order 2\"\n end if;\n\n (df2,df1) := sel ectremove(_U->has([op(_U)],diff),derivs);\n if nops(df2)<>1 or nops( df1)<>1 then \n error \"the derivatives, %1, do not make sense\", derivs;\n end if; \n (df2,df1) := (op(df2),op(df1));\n\n # Get t he arguments in the derivatives.\n if type(df1,function) and op(0,df 1)=diff and nops(df1)=2 then\n yx := op(1,df1);\n if not typ e(yx,anyfunc(name)) then\n error \"the 1st argument %1, in the derivative, %2, is invalid .. it should be the 'unknown' dependent va riable\",yx,df1;\n end if; \n x := op(2,df1);\n if not \+ type(x,name) then\n error \"the 2nd argument %1, in the deriva tive, %2, is invalid .. it should be the independent variable\",x,df1; \n end if; \n else\n error \"the derivative, %1, does not \+ make sense\",df1;\n end if;\n\n if type(df2,function) and nops(df2 )=2 and op(0,df2)='diff' then\n (df3,x3) := selectremove(has,\{op (df2)\},diff);\n if nops(df3)<>1 or nops(x3)<>1 then \n e rror \"the derivative, %1, does not make sense\",df2;\n end if;\n (df3,x3) := (op(df3),op(x3));\n if type(df3,function) and n ops(df3)=2 and op(0,df3)='diff' then\n yx2 := op(1,df3);\n \+ if not type(yx2,anyfunc(name)) then\n error \"the 1st \+ argument %1, in the derivative, %2, is invalid .. it should be the 'un known' dependent variable\",yx2,df3;\n end if; \n x2 : = op(2,df2);\n if not type(x2,name) then\n error \" the 2nd argument %1, in the derivative, %2, is invalid .. it should be the independent variable\",x2,df3;\n end if; \n if no t x2=x3 then\n error \"the 2nd arguments, %1 and %2 in the \+ derivatives %3 and %4 should be the same\",x2,x3,df2,df3;\n en d if;\n else\n error \"the derivative, %1, does not make \+ sense\",df3;\n end if\n else\n error \"the derivative, %1, does not make sense\",df2;\n end if;\n\n # Arguments in the 2 der ivatives must be the same.\n if x2<>x or yx2<>yx then\n error \+ \"the differential equation contains inconsistent arguments\"\n end \+ if;\n\n y := op(0,yx);\n vars := indets(de,name);\n if member(y, vars) then\n error \"%1 and %2 cannot both appear in the differen tial equation\",yx,y;\n end if;\n if op(1,yx)<>x then\n error \"the derivatives do not make sense\"\n end if;\n\n if startopts= 3 then \n if x<>xx or y<>yy then\n error \"cannot solve t he differential equation for %1\",ee;\n end if;\n end if;\n \+ \n if assigned(C) and not type(eval(C),table) then\n C := table ();\n WARNING(\"C has been redefined as a table for use as arbitr ary constants\");\n end if;\n # Find the indices j1,j2 to use in t he constants\n Cs := select(type,indets(de),'specindex(posint,C)'); \n gt := proc(_u) local s,j;\n typematch(_u,C[j::posint], 's'); \n subs(s,j)\n end proc:\n jS := sort([op(m ap(gt,Cs))]);\n for i to nops(jS)+1 do\n if not member(i,jS) th en j1 := i; break; end if; \n end do;\n jS := sort([j1,op(jS)]); \n for i to nops(jS)+1 do\n if not member(i,jS) then j2 := i; b reak; end if; \n end do;\n\n # Form a polynomial by substituting for the derivatives.\023\n pol:= subs(yx=d0,subs(diff(yx,x)=d1,\n \+ subs(diff(yx,x$2)=d2,de)));\n pol := d2-expand(rhs(i solate(pol,d2)));\n\n if not type(pol,polynom(anything,[d0,d1,d2])) \+ then\n error \"the DE is not linear\"\n end if;\n \n if degre e(pol,d2)<>1 or not member(degree(pol,d1),\{0,1\})\n \+ or not member(degree(pol,d0),\{0,1\}) then\n error \"the DE \+ is not linear\"\n end if;\n \n # Coefficients of DE are polynomi al coefficients.\n c2 := traperror(coeff(pol,d1,1));\n if c2=laste rror or member(d0,indets(c2,name)) or\n member(d2,indets(c2,name) ) then \n error \"the DE is not linear\"\n end if;\n c1 := tr aperror(coeff(pol,d0,1));\n if c1=lasterror or member(d1,indets(c1,n ame))\n or member(d2,indets(c1,name)) then\n error \"the DE \+ is not linear\"\n end if;\n c0 := traperror(coeff(pol,d1,0));\n \+ if c0=lasterror then error \"the DE is not linear\" end if;\n c0 := \+ simplify(d2+c1*d0-c0);\n if member(d0,indets(c0,name)) or member(d2, indets(c0,name))\n then error \"the DE is not linear\"\n end if ;\n c2 := simplify(c2);\n c1 := simplify(c1);\n \n if member(x,i ndets(c2,name)) or member(x,indets(c1,name)) then\n error \"the c oefficients must be independent of the variable %1\",x;\n end if;\n \+ \n numcoeff := evalb(type(c1,realcons) and type(c2,realcons));\n \+ homog := evalb(c0=0);\n\n if initcond then\n # Get the boundar y values.\n lsic := lhs(ic0);\n if type(lsic,function) and o p(0,lsic)=y and nops(lsic)=1 \n and type(o p(1,lsic),algebraic) then\n x0 := op(1,lsic);\n if has (x0,\{x,y\}) then\n error \"the boundary conditions must no t involve %1 or %2\",x,y;\n end if;\n y0 := rhs(ic0); \n t0 := 0; # flag for y coord or derivative\n if has( y0,\{x,y\}) then\n error \"the boundary conditions must not involve %1 or %2\",x,y;\n end if;\n elif type(lsic,funct ion) and op(0,lsic)=D(y) and nops(lsic)=1 \n \+ and type(op(1,lsic),algebraic) then\n x0 := op(1,lsic);\n \+ if has(x0,\{x,y\}) then\n error \"the boundary condi tions must not involve %1 or %2\",x,y;\n end if;\n y0 \+ := rhs(ic0);\n t0 := 1; # flag for y coord or derivative\n \+ if has(y0,\{x,y\}) then\n error \"the boundary conditi ons must not involve %1 or %2\",x,y;\n end if;\n else\n \+ error \"boundary condition is not decipherable\"\n end if; \n\n lsic := lhs(ic1);\n if type(lsic,function) and op(0,lsi c)=y and nops(lsic)=1 \n and type(op(1,lsi c),algebraic) then\n x1 := op(1,lsic);\n if has(x1,\{x ,y\}) then\n error \"the boundary conditions must not invol ve %1 or %2\",x,y;\n end if;\n y1 := rhs(ic1);\n \+ t1 := 0; # flag for y coord or derivative\n if has(y1,\{x,y \}) then\n error \"the boundary conditions must not involve %1 or %2\",x,y;\n end if;\n elif type(lsic,function) and op(0,lsic)=D(y) and nops(lsic)=1 \n and t ype(op(1,lsic),algebraic) then\n x1 := op(1,lsic);\n i f has(x1,\{x,y\}) then\n error \"the boundary conditions mu st not involve %1 or %2\",x,y;\n end if;\n y1 := rhs(i c1);\n t1 := 1; # flag for y coord or derivative\n if \+ has(y1,\{x,y\}) then\n error \"the boundary conditions must not involve %1 or %2\",x,y;\n end if;\n else\n e rror \"boundary condition is not decipherable\"\n end if;\n\n \+ if t0=t1 and x0=x1 then\n error \"impossible boundary condit ions\"\n end if;\n if signm(x1-x0)=-1 then # swap over\n \+ tt := x1; x1 := x0; x0 := tt;\n tt := y1; y1 := y0; y0 := tt;\n tt := t1; t1 := t0; t0 := tt;\n end if;\n end if ;\n\n e3 := algfact([c1,c2],x);\n e2 := simplify(c2*e3);\n e1 := simplify(c1*e3);\n\n cd := ratfact([e1,e2,e3]);\n e3 := simplify( e3*cd);\n e2 := simplify(e2*cd);\n e1 := simplify(e1*cd);\n if p rntflg then\n print(`auxiliary equation . . `,e3*m^2+e2*m+e1=0);p rint(``);\n end if;\n\n bb := simplify(-c2/2);\n bs := bb^2;\n \+ if numcoeff then\n rootstype := signm(bs-c1);\n eqrts := ro otstype=0;\n realrts := rootstype=1;\n cmplxrts := rootstype =-1; \n if realrts then\n dsc := sqrt(bs-c1);\n m 1 := simplify(bb+dsc);\n m2 := simplify(bb-dsc);\n if \+ prntflg then\n print(`roots . . `,m1,m2);print(``);\n \+ end if;\n h1x := exp(m1*x);\n h2x := exp(m2*x);\n \+ cx := C[j1]*h1x+C[j2]*h2x;\n if homog then\n \+ soln := yx=cx;\n else\n if prntflg then\n \+ print(`complementary solution . . `,cx);print(``);\n \+ end if;\n tx := c0*exp(c2*x)/(2*dsc);\n v1x:= si mplify(exp(m2*x)*tx);\n temp := int(v1x,x);\n u1 OK := indets(temp,'specfunc(anything,int)')=\{\};\n if u1O K then\n u1x := temp;\n else\n \+ u1x := Intat(subs(x=_u,v1x),_u=x);\n end if;\n v 2x:= simplify(-exp(m1*x)*tx);\n temp := int(v2x,x);\n \+ u2OK := indets(temp,'specfunc(anything,int)')=\{\};\n \+ if u2OK then\n u2x := temp;\n else\n \+ u2x := Intat(subs(x=_u,v2x),_u=x);\n end if;\n \+ px := rationalize(simplify(h1x*u1x+h2x*u2x));\n if p rntflg then\n print(`particular solution . . `,h1x*Int(v 1x,x)+h2x*Int(v2x,x));\n print(` =`,px);print(``); \n end if;\n soln := yx=px+cx;\n end if; \n elif cmplxrts then\n q := sqrt(c1-bs);\n m1 := bb + q*Complex(1);\n m2 := bb - q*Complex(1);\n if pr ntflg then\n print(`roots . . `,m1,m2);print(``);\n \+ end if;\n h1x := exp(bb*x)*sin(q*x);\n h2x := exp(bb* x)*cos(q*x);\n cx :=C[j1]*h1x+C[j2]*h2x;\n if homog th en\n soln := yx=cx;\n else\n if prntflg \+ then\n print(`complementary solution . . `,cx);print(``) ;\n end if;\n tx := c0/(q*exp(bb*x));\n \+ v1x:= combine(simplify(cos(q*x)*tx),trig);\n temp := int (v1x,x);\n u1OK := indets(temp,'specfunc(anything,int)')= \{\};\n if u1OK then\n u1x := temp;\n \+ else\n u1x := Intat(subs(x=_u,v1x),_u=x);\n \+ end if;\n v2x:= combine(simplify(-sin(q*x)*tx),trig);\n \+ temp := int(v2x,x);\n u2OK := indets(temp,'spec func(anything,int)')=\{\};\n if u2OK then\n u 2x := temp;\n else\n u2x := Intat(subs(x=_u,v 2x),_u=x);\n end if;\n px := rationalize(combine (simplify(h1x*u1x+h2x*u2x),trig));\n if prntflg then\n \+ print(`particular solution . . `,h1x*Int(v1x,x)+h2x*Int(v2x, x));\n print(` =`,px);print(``);\n end \+ if;\n soln := yx=px+cx;\n end if;\n else\n \+ if prntflg then\n print(`single root . . `,bb);\n \+ end if;\n h1x := exp(bb*x);\n h2x := x*exp(bb*x); \n cx := C[j1]*h1x+C[j2]*h2x;\n if homog then\n \+ soln :=yx=cx;\n else\n if prntflg then\n \+ print(`complementary solution . . `,cx);print(``);\n \+ end if;\n v2x := simplify(c0/exp(bb*x));\n v1 x := simplify(-x*v2x);\n temp := int(v1x,x);\n u 1OK := indets(temp,'specfunc(anything,int)')=\{\};\n if u1 OK then\n u1x := temp;\n else\n \+ u1x := Intat(subs(x=_u,v1x),_u=x);\n end if;\n \+ temp := int(v2x,x);\n u2OK := indets(temp,'specfunc(anythi ng,int)')=\{\};\n if u2OK then\n u2x := temp; \n else\n u2x := Intat(subs(x=_u,v2x),_u=x); \n end if;\n px := rationalize(combine(simplify (h1x*u1x+h2x*u2x)));\n if prntflg then\n prin t(`particular solution . . `,h1x*Int(v1x,x)+h2x*Int(v2x,x));\n \+ print(` =`,px);print(``);\n end if;\n \+ soln := yx=px+cx;\n end if;\n end if;\n else # non-n umeric coefficients\n dscsq := normal(e2^2-4*e3*e1);\n dsc : = sqrt(dscsq,symbolic);\n cmplxrts := signum(dscsq)=-1 or hastyp e(dsc,nonreal);\n m1 := simplify(-1/2*(e2+dsc)/e3);\n m2 := \+ simplify(-1/2*(e2-dsc)/e3);\n eqrts := evalb(m1=m2);\n realr ts := not (cmplxrts or eqrts);\n if realrts then\n if prn tflg then\n print(`roots . . `,m1,m2);\n end if;\n \+ h1x := exp(m1*x);\n h2x := exp(m2*x);\n \+ cx := C[j1]*h1x+C[j2]*h2x;\n if homog then\n sol n := yx=cx;\n else\n if prntflg then\n \+ print(`complementary solution . . `,cx);print(``);\n end \+ if;\n v1x := simplify(-c0/exp(m1*x));\n temp := \+ int(v1x,x);\n u1OK := indets(temp,'specfunc(anything,int)' )=\{\};\n if u1OK then\n u1x := temp;\n \+ else\n u1x := Intat(subs(x=_u,v1x),_u=x);\n \+ end if;\n v2x:= simplify(c0/exp(m2*x));\n t emp := int(v2x,x);\n u2OK := indets(temp,'specfunc(anythin g,int)')=\{\};\n if u2OK then\n u2x := temp; \n else\n u2x := Intat(subs(x=_u,v2x),_u=x); \n end if;\n px := rationalize(simplify(e3/dsc*( h1x*u1x+h2x*u2x)));\n if prntflg then\n print (`particular solution . . `,h1x*Int(v1x,x)+h2x*Int(v2x,x));\n \+ print(` =`,px);print(``);\n end if;\n \+ soln := yx=px+cx;\n end if;\n elif cmplxrts then\n \+ q := simplify(1/2*dsc/Complex(1)/e3);\n if hastype(q,nonre al) then\n q := sqrt(-dscsq,symbolic)\n end if;\n \+ m1 := bb + q*Complex(1);\n m2 := bb - q*Complex(1);\n \+ if prntflg then\n print(`roots . . `,m1,m2);print(`` );\n end if;\n h1x := exp(bb*x)*sin(q*x);\n h2 x := exp(bb*x)*cos(q*x);\n cx := C[j1]*h1x+C[j2]*h2x;\n \+ if homog then\n soln := yx=cx;\n else\n \+ if prntflg then\n print(`complementary solution . . `, cx);print(``);\n end if;\n tx := c0/(q*exp(bb*x) );\n v1x:= combine(simplify(cos(q*x)*tx),trig);\n \+ temp := int(v1x,x);\n u1OK := indets(temp,'specfunc(anyt hing,int)')=\{\};\n if u1OK then\n u1x := tem p;\n else\n u1x := Intat(subs(x=_u,v1x),_u=x) ;\n end if;\n v2x := combine(simplify(-sin(q*x)* tx),trig);\n temp := int(v2x,x);\n u2OK := inde ts(temp,'specfunc(anything,int)')=\{\};\n if u2OK then\n \+ u2x := temp;\n else\n u2x := Inta t(subs(x=_u,v2x),_u=x);\n end if;\n px := ration alize(combine(h1x*u1x+h2x*u2x,trig));\n if prntflg then\n \+ print(`particular solution . . `,h1x*Int(v1x,x)+h2x*Int(v 2x,x));\n print(` =`,simplify(px));print(``);\n \+ end if;\n soln := yx=px+cx;\n end if;\n \+ else\n if prntflg then\n print(`single root . . \+ `,bb);\n end if;\n h1x := exp(bb*x);\n h2x := \+ x*exp(bb*x);\n cx := C[j1]*h1x+C[j2]*h2x;\n if homog t hen\n soln :=yx=cx;\n else\n if prntflg \+ then\n print(`complementary solution . . `,cx);print(``) ;\n end if;\n v2x := simplify(c0/exp(bb*x));\n \+ v1x := simplify(-x*v2x);\n temp := int(v1x,x);\n \+ u1OK := indets(temp,'specfunc(anything,int)')=\{\};\n \+ if u1OK then\n u1x := temp;\n else\n \+ u1x := Intat(subs(x=_u,v1x),_u=x);\n end if;\n temp := int(v2x,x);\n u2OK := indets(temp,'spe cfunc(anything,int)')=\{\};\n if u2OK then\n \+ u2x := temp;\n else\n u2x := Intat(subs(x=_u, v2x),_u=x);\n end if;\n px := rationalize(sim plify(h1x*u1x+h2x*u2x));\n if prntflg then\n \+ print(`particular solution . . `,h1x*Int(v1x,x)+h2x*Int(v2x,x));\n \+ print(` =`,px);print(``);\n end if;\n \+ soln := yx=px+cx;\n end if;\n end if;\n end if; \n \n if initcond then\n if prntflg then\n print(`gen eral solution . . `);\n print(soln);print(``);\n end if; \n\n # derivatives of generating solutions\n if realrts then \n k1x := m1*h1x;\n k2x := m2*h2x;\n elif eqrts t hen\n k1x := bb*h1x;\n k2x := h1x+bb*h2x;\n else \n k1x := exp(bb*x)*(bb*sin(q*x)+q*cos(q*x));\n k2x := exp(bb*x)*(bb*cos(q*x)-q*sin(q*x));\n end if;\n \n # d erivative of complementary solution\n dcx := C[j1]*k1x+C[j2]*k2x; \n \n # derivative of particular solution\n if not homo g then\n dpx := u1x*k1x+u2x*k2x;\n end if;\n\n if t0 =0 then\n if homog then\n yy := simplify(eval(subs( x=x0,cx)));\n else\n if u1OK and u2OK then\n \+ yy := simplify(eval(subs(x=x0,px+cx)));\n elif u1OK then\n uhx := simplify(eval(subs(x=x0,u1x*h1x)));\n \+ hx := simplify(eval(subs(x=x0,h2x)));\n yy := uhx+hx*Intat(subs(x=_u,v2x),_u=x0)+\n simplify( eval(subs(x=x0,cx)));\n elif u2OK then\n hx : = simplify(eval(subs(x=x0,h1x)));\n uhx := simplify(eval (subs(x=x0,u2x*h2x)));\n yy := hx*Intat(subs(x=_u,v1x),_ u=x0)+uhx+\n simplify(eval(subs(x=x0,cx)));\n \+ else\n hx := simplify(eval(subs(x=x0,h1x)));\n \+ uhx := simplify(eval(subs(x=x0,h2x)));\n y y := hx*Intat(subs(x=_u,v1x),_u=x0)+\n uhx*Intat(s ubs(x=_u,v2x),_u=x0)+\n simplify(eval(subs(x=x0, cx)));\n end if;\n end if;\n else\n if homog then\n yy := simplify(eval(subs(x=x0,dcx)));\n \+ else\n if u1OK and u2OK then\n yy := s implify(eval(subs(x=x0,dpx+dcx)));\n elif u1OK then\n \+ uhx := simplify(eval(subs(x=x0,u1x*k1x)));\n hx := simplify(eval(subs(x=x0,k2x)));\n yy := uhx+hx*Intat (subs(x=_u,v2x),_u=x0)+\n simplify(eval(subs(x=x 0,dcx)));\n elif u2OK then\n hx := simplify(e val(subs(x=x0,k1x)));\n uhx := simplify(eval(subs(x=x0,u 2x*k2x)));\n yy := hx*Intat(subs(x=_u,v1x),_u=x0)+uhx+\n simplify(eval(subs(x=x0,dcx)));\n el se\n hx := simplify(eval(subs(x=x0,k1x)));\n \+ uhx := simplify(eval(subs(x=x0,k2x)));\n yy := hx*Int at(subs(x=_u,v1x),_u=x0)+\n uhx*Intat(subs(x=_u,v2 x),_u=x0)+\n simplify(eval(subs(x=x0,dcx)));\n \+ end if;\n end if;\n end if;\n eq0 := y0=yy ;\n\n if t1=0 then\n if homog then\n yy := sim plify(eval(subs(x=x1,cx)));\n else\n if u1OK and u2 OK then\n yy := simplify(eval(subs(x=x1,px+cx)));\n \+ elif u1OK then\n uhx := simplify(eval(subs(x=x1,u 1x*h1x)));\n hx := simplify(subs(x=x1,h2x));\n \+ yy := uhx+hx*Intat(subs(x=_u,v2x),_u=x1)+\n \+ simplify(eval(subs(x=x1,cx)));\n elif u2OK then\n \+ hx := simplify(eval(subs(x=x1,h1x)));\n uhx := sim plify(eval(subs(x=x1,u2x*h2x)));\n yy := hx*Intat(subs(x =_u,v1x),_u=x1)+uhx+\n simplify(eval(subs(x=x1,c x)));\n else\n hx := simplify(eval(subs(x=x1, h1x)));\n uhx := simplify(eval(subs(x=x1,h2x)));\n \+ yy := hx*Intat(subs(x=_u,v1x),_u=x1)+\n u hx*Intat(subs(x=_u,v2x),_u=x1)+\n simplify(eval( subs(x=x1,cx)));\n end if;\n end if;\n else\n \+ if homog then\n yy := simplify(eval(subs(x=x1,dcx)) );\n else\n if u1OK and u2OK then\n y y := simplify(eval(subs(x=x1,dpx+dcx)));\n elif u1OK then\n uhx := simplify(eval(subs(x=x1,u1x*k1x)));\n \+ hx := simplify(eval(subs(x=x1,k2x)));\n yy := uhx+hx *Intat(subs(x=_u,v2x),_u=x1)+\n simplify(eval(su bs(x=x1,dcx)));\n elif u2OK then\n hx := simp lify(eval(subs(x=x1,k1x)));\n uhx := simplify(eval(subs( x=x1,u2x*k2x)));\n yy := hx*Intat(subs(x=_u,v1x),_u=x1)+ uhx+\n simplify(eval(subs(x=x1,dcx)));\n \+ else\n hx := simplify(eval(subs(x=x1,k1x)));\n \+ uhx := simplify(eval(subs(x=x1,k2x)));\n yy := \+ hx*Intat(subs(x=_u,v1x),_u=x1)+\n uhx*Intat(subs(x =_u,v2x),_u=x1)+\n simplify(eval(subs(x=x1,dcx)) );\n end if;\n end if;\n end if;\n eq1 := y1=yy;\n\n if prntflg then\n print(`from the initial con ditions . . `);\n print(eq0);print(eq1);print(``);\n end \+ if;\n cvals := traperror(simplify(solve(\{eq0,eq1\},\{C[j1],C[j2] \})));\n if cvals=lasterror then\n error \"cannot determi ne the constants %1 and %2\",C[j1],C[j2];\n end if;\n const1 := subs(cvals,C[j1]);\n const2 := subs(cvals,C[j2]);\n if p rntflg then\n print(`so that . . `);\n print(C[j1] = c onst1);\n print(C[j2] = const2);print(``);\n end if;\n \+ soln := subs(\{C[j1] = const1,C[j2] = const2\},soln);\n else\n \+ if not homog and not eqrts then\n p2x := frontend(expand,[p x]);\n p2x := map(simplify,p2x);\n if numcoeff then\n \+ if op(0,p2x)=`+` then\n sm := 0;\n \+ omit := 0; \n for i from 1 to nops(p2x) do\n \+ term := op(i,p2x);\n if not (patmatch(term, a::complexcons*h1x)\n or patmatch(term,a::complex cons*h2x)) then\n sm := sm + term;\n \+ else\n omit := omit + term;\n \+ end if;\n end do;\n if p2x<>sm then\n \+ px := sm;\n if prntflg then\n \+ print(omit, ` in the particular solution can be absorbed in to the complementary solution`);\n print(``);\n \+ end if;\n end if;\n end if;\n \+ else # non-numeric coefficients \n if op(0,p2x)=`+` then\n sm := 0;\n omit := 0; \n \+ for i from 1 to nops(p2x) do\n term := simplify( op(i,p2x));\n addterm := true;\n h1x := simplify(h1x);\n if patmatch(term,a::algebraic*h1 x,'la') then\n if not has(subs(la,a),x) then\n \+ addterm := false;\n end if;\n \+ end if;\n h2x := simplify(h2x);\n \+ if patmatch(term,a::algebraic*h2x,'la') then\n \+ if not has(subs(la,a),x) then\n add term := false;\n end if;\n end if ;\n if addterm then\n sm := sm + \+ term;\n else\n omit := omit + ter m;\n end if; \n end do;\n \+ if p2x<>sm then\n px := sm;\n if pr ntflg then\n print(omit, ` in the particular solut ion can be absorbed into the complementary solution`);\n \+ print(``);\n end if;\n end if;\n end if;\n end if;\n soln := yx=simplify(px )+cx;\n end if;\n end if;\n return soln;\nend proc:" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "Examples are given in \+ the following sections." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveCC" }{TEXT -1 27 ": general solution examples" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 108 "These are the same examples as were solved by the method of undet ermined coefficients in another worksheet. " }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1 " }}{PARA 257 "" 0 "" {TEXT -1 6 " 2 " }{XPPEDIT 18 0 "d^2*y/(d*x^2 )+dy/dx-y = x^2+3;" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\" \"F)*&%#dyGF)%#dxGF-F)F(F-,&*$F,F'F)\"\"$F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "de := 2*diff(y(x),x$2)+diff(y(x),x)-y(x)=x^2+3;\ndesolveCC(de,y(x),info= true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\"\"#\"\"\"-%%dif fG6$-%\"yG6#%\"xG-%\"$G6$F0F(F)F)-F+6$F-F0F)F-!\"\",&*$)F0F(F)F)\"\"$F )" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,(*&\" \"#\"\"\")%\"mGF'F(F(F*F(F(!\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G#\"\"\"\"\"#!\" \"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 257 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2" "6#,&*(%\" dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx-y = 2*x-3;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF ),&*&\"\"#F'%\"xGF'F'\"\"$F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "diff(y(x),x$2)+2*d iff(y(x),x)-y(x)=2*x-3;\ndesolve(%,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F+\"\"#\"\"\"*&F/F0-F &6$F(F+F0F0F(!\"\",&*&F/F0F+F0F0\"\"$F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,(*$)%\"mG\"\"#\"\"\"F**&F)F*F(F*F*F* !\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G,&*$\"\"##\"\"\"F&F(F(!\"\",&F(F)F%F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 257 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\" \"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/d x+4*y = 6*exp(-x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'%\"yGF'F'* &\"\"'F'-%$expG6#,$%\"xGF)F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "de := diff(y(x),x$ 2)+2*diff(y(x),x)+4*y(x)=6*exp(-x);\ndesolveCC(de,y(x),info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G 6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2F2*&\"\"%F2F*F2F2,$*&\"\"'F2-%$expG6#, $F-!\"\"F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~. ~.~G/,(*$)%\"mG\"\"#\"\"\"F**&F)F*F(F*F*\"\"%F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G ,&\"\"\"!\"\"*&\"\"$#F%\"\"#^#F%F%F%,&F%F&*&^#F&F%F(F)F%" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 257 "" 0 "" {TEXT -1 4 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)-4*y = 4*sin(2*x)-4*cos(2*x);" "6#/,&*(%\" dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&\"\"%F)F(F)F-,&*&F/F)-%$si nG6#*&F'F)F,F)F)F)*&F/F)-%$cosG6#*&F'F)F,F)F)F-" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "de := diff(y(x),x$2)-4*y(x)=4*sin(2*x)-4*cos(2*x);\ndesolveCC(de,i nfo=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"y G6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2!\"\",&*&F4F2-%$sinG6#,$*&F1 F2F-F2F2F2F2*&F4F2-%$cosGF:F2F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8a uxiliary~equation~.~.~G/,&*$)%\"mG\"\"#\"\"\"F*\"\"%!\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%+roots~.~.~G\"\"#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-5;" "6#,&*( %\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"&F," }{TEXT -1 2 " \+ " }{XPPEDIT 18 0 "dy/dx+4*y = 8*exp(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\" F'*&\"\"%F'%\"yGF'F'*&\"\")F'-%$expG6#%\"xGF'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "de := diff(y(x),x$2)-5*diff(y(x),x)+4*y(x)=8*exp(x);\ndesolveCC(de ,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-% \"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"&F2-F(6$F*F-F2!\"\"*&\"\"%F2F*F2 F2,$*&\"\")F2-%$expGF,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxili ary~equation~.~.~G/,(*$)%\"mG\"\"#\"\"\"F**&\"\"&F*F(F*!\"\"\"\"%F*\" \"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G\"\"%\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+4*y = 4*sin(2*x)-4*cos( 2*x);" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&\"\"%F)F (F)F),&*&F/F)-%$sinG6#*&F'F)F,F)F)F)*&F/F)-%$cosG6#*&F'F)F,F)F)F-" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "de := diff(y(x),x$2)+4*y(x)=4*sin(2*x)-4*cos(2*x );\ndesolveCC(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG /,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2F*F2F2,&*&F4F2 -%$sinG6#,$*&F1F2F-F2F2F2F2*&F4F2-%$cosGF9F2!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,&*$)%\"mG\"\"#\"\"\"F*\"\"% F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G^#\"\"#^#!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,,*&-%$sinG6#,$F'\"\"#\"\"\"%$_C2GF/F/*&-%$cosGF,F /%$_C1GF/F/*&#F/F.F/F2F/!\"\"*&F*F/F'F/F7*&F2F/F'F/F7" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\"\"#% \"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F," }{TEXT -1 3 " " }{XPPEDIT 18 0 "dy/dx+y = exp(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'-%$expG 6#%\"xG" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "de := diff(y(x),x$2)-2*diff(y(x),x) +y(x)=exp(x);\ndesolveCC(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$ F*F-F2!\"\"F*F2-%$expGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliar y~equation~.~.~G/,(*$)%\"mG\"\"#\"\"\"F**&F)F*F(F*!\"\"F*F*\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $%1single~root~.~.~G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveCC" }{TEXT -1 30 ": particular solution examples" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 1" }}{PARA 257 "" 0 "" {TEXT -1 3 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+y=x^3" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F )*$%\"xGF'F)!\"\"F)F(F)*$F,\"\"$" }{TEXT -1 4 ", " }{XPPEDIT 18 0 "` y '`(1) = 2;" "6#/-%$y~'G6#\"\"\"\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(3 )= 1" "6#/-%\"yG6#\"\"$\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "de : = diff(y(x),x$2)+y(x) = x^3;\nic := D(y)(1)=2,y(3)=1;\ndesolveCC(\{de, ic\},y(x),info=true);\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #deG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"F*F2*$)F-\"\"$F2" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/--%\"DG6#%\"yG6#\"\"\"\"\"#/ -F+6#\"\"$F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~. ~.~G/,&*$)%\"mG\"\"#\"\"\"F*F*F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G^#\"\"\"^#!\"\" " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%F*F0*&F=F*F5F*F**(F9F*F>F*FBF*F*F*F@F0-F6F&F*F0" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&F'\"\"\",&*$)F'\"\"# F*F*$\"\"'\"\"!!\"\"F*F**&$\"+Du62G!\")F*-%$sinGF&F*F**&$\"+==B37F6F*- %$cosGF&F*F*" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dsolve(\{de, ic\},y(x));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\" xG,(*(-%$sinGF&\"\"\",(*&\"\"&F,-%$cosG6#F,F,!\"\"*(\"#?F,F0F,)-F+F2\" \"#F,F,*&\"\")F,F7F,F,F,,&F,F3*&F8F,F6F,F,F3F,*(-F1F&F,,(*&\"#:F,F7F,F 3*&F5F,)F7\"\"$F,F,*&F:F,F0F,F3F,F;F3F3*&F'F,,&*$)F'F8F,F,\"\"'F3F,F, " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&$\"+@u62G!\")\" \"\"-%$sinGF&F-F-*&$\"+;=B37F,F--%$cosGF&F-F-*&F'F-,&*$)F'\"\"#F-F-$\" \"'\"\"!!\"\"F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Ex ample 2" }}{PARA 257 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^2*y/( d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"$F( " }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = 0;" "6#/,&*&%#dyG\"\"\"% #dxG!\"\"F'*&\"\"#F'%\"yGF'F'\"\"!" }{TEXT -1 3 ", " }{XPPEDIT 18 0 " y(1) = 2" "6#/-%\"yG6#\"\"\"\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (3) = 1" "6#/-%\"yG6#\"\"$\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "de := diff(y (x),x$2)+3*diff(y(x),x)+2*y(x) = 0;\nic := y(1)=2,y(3)=1;\ndesolveCC( \{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6 $-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"$F2-F(6$F*F-F2F2*&F1F2F*F2F2 \"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"\"\"\"#/ -F(6#\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&**,&-% $expG6#\"\"%\"\"\"\"\"#!\"\"F/-F,6#F/F/,&-F,6#F0F/F/F1F1-F,6#,$F'F1F/F /**,&F0F1F5F/F/F+F/F4F1-F,6#,$*&F0F/F'F/F1F/F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de, ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&**,&-%$ expG6#\"\"%\"\"\"\"\"#!\"\"F/-F,6#F/F/,&-F,6#F0F/F/F1F1-F,6#,$F'F1F/F/ **,&F0F1F5F/F/F+F/F4F1-F,6#,$*&F0F/F'F/F1F/F1" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\" *&F%F(*$%\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx +2*y = sin(3*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'-%$ sinG6#*&\"\"$F'%\"xGF'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(1) = 2 ;" "6#/-%$y~'G6#\"\"\"\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y(3) = \+ 1" "6#/-%\"yG6#\"\"$\"\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "de := diff(y(x),x $2) + 3*diff(y(x),x) + 2*y(x) = sin(3*x);\nic := D(y)(1)=2,y(3)=1;\nde solveCC(\{de,ic\},y(x)):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"$F2-F(6$F* F-F2F2*&F1F2F*F2F2-%$sinG6#,$*&F4F2F-F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/--%\"DG6#%\"yG6#\"\"\"\"\"#/-F+6#\"\"$F-" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&$\"+Bp2Bp!#6\"\"\"-% $cosG6#,$*&$\"\"$\"\"!F-F'F-F-F-!\"\"*&$\"+&Q:YQ&F,F--%$sinGF0F-F6*&$ \"+v@(>5#!\")F--%$expG6#,$*&$F-F5F-F'F-F6F-F-*&$\"+>*pe_$F?F--FA6#,$*& $\"\"#F5F-F'F-F6F-F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dsolve(\{de,ic\},y(x)):\nevalf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&$\"+Bp2Bp!#6\"\"\"-% $cosG6#,$*&$\"\"$\"\"!F-F'F-F-F-!\"\"*&$\"+&Q:YQ&F,F--%$sinGF0F-F6*&$ \"+v@(>5#!\")F--%$expG6#,$*&$F-F5F-F'F-F6F-F-*&$\"+>*pe_$F?F--FA6#,$*& $\"\"#F5F-F'F-F6F-F6" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y /(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"% F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+13*y = 3*exp(x);" "6#/,&*&%# dyG\"\"\"%#dxG!\"\"F'*&\"#8F'%\"yGF'F'*&\"\"$F'-%$expG6#%\"xGF'" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "y (1) = 2" "6#/-%\"yG6#\"\"\"\"\"#" } {TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(3) = 1;" "6#/-%$y~'G6#\"\"$\"\" \"" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "de := diff(y(x),x$2) + 4*diff(y(x),x) + \+ 13*y(x) = 3*exp(x);\nic := y(1)=2,D(y)(3)=1;\ndesolveCC(\{de,ic\},y(x) ):\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-% \"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"%F2-F(6$F*F-F2F2*&\"#8F2F*F2F2,$ *&\"\"$F2-%$expGF,F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-% \"yG6#\"\"\"\"\"#/--%\"DG6#F(6#\"\"$F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&$\"+nmmm;!#5\"\"\"-%$expGF&F-F-*($\"+h;l1F!\"( F--F/6#,$*&$\"\"#\"\"!F-F'F-!\"\"F--%$sinG6#,$*&$\"\"$F:F-F'F-F-F-F-*( $\"+n8j.F!\")F-F4F--%$cosGF>F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dsolve(\{de,ic\},y(x)):\neva lf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*($\"+h;l1F! \"(\"\"\"-%$expG6#,$*&$\"\"#\"\"!F-F'F-!\"\"F--%$sinG6#,$*&$\"\"$F5F-F 'F-F-F-F-*($\"+r8j.F!\")F-F.F--%$cosGF9F-F-*&$\"+nmmm;!#5F--F/F&F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+9*y = 5*sin( 3*x);" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&\"\"*F)F (F)F)*&\"\"&F)-%$sinG6#*&\"\"$F)F,F)F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (0) = 2" "6#/-%\"yG6#\"\"!\"\"#" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 0;" "6#/-%$y~'G6#\"\"!F'" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "de := diff(y(x),x$2)+9*y(x)=5*sin(3*x);\nic := y(0)=2,D(y)(0)=0;\ndesolv eCC(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%di ffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&\"\"*F2F*F2F2,$*&\"\"&F2-%$si nG6#,$*&\"\"$F2F-F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/ -%\"yG6#\"\"!\"\"#/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /-%\"yG6#%\"xG,(*&#\"\"&\"#=\"\"\"-%$sinG6#,$*&\"\"$F-F'F-F-F-F-*&#F+ \"\"'F-*&-%$cosGF0F-F'F-F-!\"\"*&\"\"#F-F8F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de, ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\" &\"#=\"\"\"-%$sinG6#,$*&\"\"$F-F'F-F-F-F-*&#F+\"\"'F-*&-%$cosGF0F-F'F- F-!\"\"*&\"\"#F-F8F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(! \"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+10*y = 5*exp(-x)*sin (3*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"#5F'%\"yGF'F'*(\"\"&F'-%$ex pG6#,$%\"xGF)F'-%$sinG6#*&\"\"$F'F3F'F'" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (0) = 1" "6#/-%\"yG6#\"\"!\"\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "`y '`(0) = 1;" "6#/-%$y~'G6#\"\"!\"\"\"" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "de := diff(y(x),x$2)+2*diff(y(x),x)+10*y(x)=5*exp(-x)*sin(3*x);\n ic := y(0)=1,D(y)(0)=1;\ndesolveCC(\{de,ic\},info=true);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"# \"\"\"*&F1F2-F(6$F*F-F2F2*&\"#5F2F*F2F2,$*(\"\"&F2-%$expG6#,$F-!\"\"F2 -%$sinG6#,$*&\"\"$F2F-F2F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#i cG6$/-%\"yG6#\"\"!\"\"\"/--%\"DG6#F(F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,(*$)%\"mG\"\"#\"\"\"F**&F)F*F(F*F*\" #5F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G^$!\"\"\"\"$^$F%!\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Afrom~the~initial~conditions~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\"&%\"CG6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/\"\"\",(#\"\"&\"#7!\"\"*&\"\"$F$&%\"CG6#F$F$F$&F-6#\" \"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%-so~that~.~.~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&% \"CG6#\"\"\"#\"#H\"#O" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\" #\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"#<\"#=\"\"\"*&-%$expG6#,$F'!\"\"F- -%$sinG6#,$*&\"\"$F-F'F-F-F-F-F-*&#\"\"&\"\"'F-*(F/F--%$cosGF6F-F'F-F- F3*&F/F-F?F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de, ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"#< \"#=\"\"\"*&-%$expG6#,$F'!\"\"F--%$sinG6#,$*&\"\"$F-F'F-F-F-F-F-*&#\" \"&\"\"'F-*(F/F--%$cosGF6F-F'F-F-F3*&F/F-F?F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 7" }}{PARA 257 "" 0 "" {TEXT -1 3 " 4 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\" *&F%F(*$%\"xGF&F(!\"\"F(\"\"%F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx +y = 0;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'\"\"!" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "y (1) = 2" "6#/-%\"yG6#\"\"\"\"\"#" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "`y '`(1) = 1;" "6#/-%$y~'G6#\"\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "de := 4*diff(y(x),x$2)-4*diff(y(x),x)+y(x) = sqrt(x) *exp(x/2);\nic := y(1)=2,D(y)(1)=1;\ndesolveCC(\{de,ic\},y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*&\"\"%\"\"\"-%%diffG6$-%\"yG 6#%\"xG-%\"$G6$F0\"\"#F)F)*&F(F)-F+6$F-F0F)!\"\"F-F)*&F0#F)F4-%$expG6# ,$*&F4F8F0F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6# \"\"\"\"\"#/--%\"DG6#F(F)F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG 6#%\"xG,(*&#\"\"\"\"#:F+*&-%$expG6#,$*&\"\"#!\"\"F'F+F+F+)F'#\"\"&F3F+ F+F+*&#F+\"#5F+*(,&\"#?F+-F/6##F+F3F+F+-F/6##F4F3F+F.F+F+F+*&#F+\"\"'F +*&F'F+F.F+F+F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "dsolve(\{de,ic\},y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"\"\"#:F+*&-%$expG6#,$*&\"\"#!\" \"F'F+F+F+)F'#\"\"&F3F+F+F+*&#F+\"#5F+*(,&\"#?F+-F/6##F+F3F+F+-F/6##F4 F3F+F.F+F+F+*&#F+\"\"'F+*&F'F+F.F+F+F4" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 8" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$% \"xGF&F(!\"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = sqrt(x) ;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'%\"yGF'-%%sqrtG6#%\"xG" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "y (0) = 0" "6#/-%\"yG6#\"\"!F'" }{TEXT -1 3 ", \+ " }{XPPEDIT 18 0 "y(1) = 0" "6#/-%\"yG6#\"\"\"\"\"!" }{TEXT -1 3 ". \+ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "de := diff(y(x),x$2)-2* diff(y(x),x)+y(x) = sqrt(x);\nic := y(0)=0,y(1)=0;\ndesolveCC(\{de,ic \},y(x));\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%di ffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"F*F2*$F-# F2F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#icG6$/-%\"yG6#\"\"!F*/-F(6# \"\"\"F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**(\"\"$\" \"\"\"\"#!\"\"F'#F+F,F+*&#F*\"\"%F+*(%#PiGF.-%$erfG6#*$F'F.F+-%$expGF& F+F+F-*&F.F+**F'F+F3F.F4F+F8F+F+F+*&#F+F1F+**,&\"\"'F-*(F3F.-F56#F+F+- F9FCF+F+F+-F96#F-F+F8F+F'F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,**&$\"+++++:!\"*\"\"\"F'#F-\"\"#F-*($\"+)QS$H8F,F--%$erfG6 #*$F'F.F--%$expGF&F-!\"\"**$\"+b#pA'))!#5F-F'F-F3F-F7F-F-*($\"+`42%y\" F=F-F7F-F'F-F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "dsolve(\{de, ic\},y(x)):\nevalf(normal(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-% \"yG6#%\"xG,**&$\"+++++:!\"*\"\"\"F'#F-\"\"#F-*($\"+)QS$H8F,F--%$erfG6 #*$F'F.F--%$expGF&F-!\"\"**$\"+b#pA'))!#5F-F'F-F3F-F7F-F-*($\"+`42%y\" F=F-F7F-F'F-F9" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveCC" }{TEXT -1 19 ": \+ symbolic examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-``(m[1]+m[2]);" "6#,&*(% \"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%!G6#,&&%\"mG6#F(F(&F26#F &F(F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+m[1]*m[2]*y = 0;" "6#/,&* &%#dyG\"\"\"%#dxG!\"\"F'*(&%\"mG6#F'F'&F,6#\"\"#F'%\"yGF'F'\"\"!" } {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "de := diff( y(x),x$2)-(m[1]+m[2])*diff(y(x),x)+m[1]*m[2]*y(x)=0;\ndesolveCC(de,y(x ));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"x G-%\"$G6$F-\"\"#\"\"\"*&,&&%\"mG6#F2F2&F66#F1F2F2-F(6$F*F-F2!\"\"*(F5F 2F8F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*& &%\"CG6#\"\"\"F--%$expG6#*&&%\"mGF,F-F'F-F-F-*&&F+6#\"\"#F--F/6#*&&F3F 6F-F'F-F-F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 2" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2) + omega^2*y=2*sin(omega*x)" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF 'F)!\"\"F)*&%&omegaGF'F(F)F)*&F'F)-%$sinG6#*&F/F)F,F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "de := diff(y(x),x$2)+omega^2*y(x)=2*sin(omega*x);\nde solveCC(de,y(x),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/ ,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&)%&omegaGF1F2F*F2F2,$* &F1F2-%$sinG6#*&F5F2F-F2F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxi liary~equation~.~.~G/,&*$)%\"mG\"\"#\"\"\"F**$)%&omegaGF)F*F*\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%+roots~.~.~G*&%&omegaG\"\"\"^#F&F&*&^#!\"\"F&F%F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 257 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+omega^2*y = sin(omega*x)+5*cos(omega*x)" "6#/,&*(%\"dG\"\"#%\"yG\"\"\"*&F&F)*$%\"xGF'F)!\"\"F)*&%&omegaGF'F(F)F ),&-%$sinG6#*&F/F)F,F)F)*&\"\"&F)-%$cosG6#*&F/F)F,F)F)F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "de := diff(y(x),x$2)+omega^2*y(x)=sin(omega*x)+5*cos( omega*x);\ndesolveCC(de,y(x),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,&-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&)%&omegaG F1F2F*F2F2,&-%$sinG6#*&F5F2F-F2F2*&\"\"&F2-%$cosGF9F2F2" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,&*$)%\"mG\"\"#\"\"\" F**$)%&omegaGF)F*F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G*&%&omegaG\"\"\"^#F&F&*&^ #!\"\"F&F%F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$%FF+F*F>F+F+ F,F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)~~~~~~~=G,$*&#\"\"\"\"\"%F '*&,*-%$sinG6#*&%&omegaGF'%\"xGF'F'*&\"\"&F'-%$cosGF-F'F'**\"#5F'F+F'F /F'F0F'F'**\"\"#F'F3F'F/F'F0F'!\"\"F'F/!\"#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*&#\"\"\"\"\"% F&*&%&omegaG!\"#-%$sinG6#*&F)F&%\"xGF&F&F&F&*&#\"\"&F'F&*&F)F*-%$cosGF -F&F&F&%fo~in~the~particular~solution~can~be~absorbed~into~the~complem entary~solutionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&#\"\"\"\"\"#F+*(F'F+,&*&\"\"&F +-%$sinG6#*&%&omegaGF+F'F+F+F+-%$cosGF3!\"\"F+F5F8F+F+*&&%\"CG6#F+F+F1 F+F+*&&F;6#F,F+F6F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&-%$sinG6#*&%&omegaG\"\"\"F'F/F/%$_C2G F/F/*&-%$cosGF,F/%$_C1GF/F/*&#F/\"\"#F/*&,(*&\"\"&F/F2F/F/**F;F/F*F/F. F/F'F/F/*(F2F/F.F/F'F/!\"\"F/F.!\"#F/F/" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 257 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+[a+b];" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F% F(*$%\"xGF&F(!\"\"F(7#,&%\"aGF(%\"bGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+a*b*y = 2*exp(-a*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*(%\" aGF'%\"bGF'%\"yGF'F'*&\"\"#F'-%$expG6#,$*&F+F'%\"xGF'F)F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "de := diff(y(x),x$2)+(a+b)*diff(y(x),x)+a*b*y(x)=2*ex p(-a*x);\ndesolveCC(de,y(x),info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&,&%\"aGF2 %\"bGF2F2-F(6$F*F-F2F2*(F5F2F6F2F*F2F2,$*&F1F2-%$expG6#,$*&F5F2F-F2!\" \"F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/, (*$)%\"mG\"\"#\"\"\"F**&,&%\"aGF*%\"bGF*F*F(F*F**&F-F*F.F*F*\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 %%+roots~.~.~G,$%\"bG!\"\",$%\"aGF&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 $% " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&-%$expG6#,$*&%\"aG\"\"\"F'F0!\" \"F0%$_C2GF0F0*&-F+6#,$*&%\"bGF0F'F0F1F0%$_C1GF0F0**\"\"#F0F*F0F'F0,&F /F1F8F0F1F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example \+ 5" }}{PARA 257 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2 )+2*p;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F&F(%\"p GF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+[p^2-q]*y = 8*exp((sqrt(q )-p)*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&7#,&*$%\"pG\"\"#F'%\"qGF)F' %\"yGF'F'*&\"\")F'-%$expG6#*&,&-%%sqrtG6#F0F'F.F)F'%\"xGF'F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "de := diff(y(x),x$2)+2*p*diff(y(x),x)+(p^2-q)*y(x)=8* exp((sqrt(q)-p)*x);\ndesolveCC(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*( F1F2%\"pGF2-F(6$F*F-F2F2*&,&*$)F4F1F2F2%\"qG!\"\"F2F*F2F2,$*&\"\")F2-% $expG6#*&,&*$F;#F2F1F2F4F " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&-%$expG6#*&,&*$%\"qG#\"\"\"\"\"#F2% \"pG!\"\"F2F'F2F2%$_C2GF2F2*&-F+6#*&,&F4F5F/F5F2F'F2F2%$_C1GF2F2**\"\" %F2-F+6#,$*&F'F2,&F/F5F4F2F2F5F2F0#F5F3F'F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 6" }}{PARA 257 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2*a;" "6#,&*(%\"dG\"\"#%\"yG\" \"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F&F(%\"aGF(F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+a^2*y = 8*exp(-a*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\" \"F'*&%\"aG\"\"#%\"yGF'F'*&\"\")F'-%$expG6#,$*&F+F'%\"xGF'F)F'" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "de := diff(y(x),x$2)+2*a*diff(y(x),x)+a^2*y(x)=8 *exp(-a*x);\ndesolveCC(de,info=true);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2%\"aGF2- F(6$F*F-F2F2*&)F4F1F2F*F2F2,$*&\"\")F2-%$expG6#,$*&F4F2F-F2!\"\"F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,(*$)%\"m G\"\"#\"\"\"F**(F)F*%\"aGF*F(F*F**$)F,F)F*F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%1single~root~.~ .~G,$%\"aG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 7" }}{PARA 257 "" 0 "" {TEXT -1 2 " " } {XPPEDIT 18 0 "p*q" "6#*&%\"pG\"\"\"%\"qGF%" }{TEXT -1 2 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+[p*r+q*s];" "6#,&*(%\"dG\"\"#%\"yG\"\"\"* &F%F(*$%\"xGF&F(!\"\"F(7#,&*&%\"pGF(%\"rGF(F(*&%\"qGF(%\"sGF(F(F(" } {TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+r*s*y = exp(-r/q*x);" "6#/,&*&%#d yG\"\"\"%#dxG!\"\"F'*(%\"rGF'%\"sGF'%\"yGF'F'-%$expG6#,$*(F+F'%\"qGF)% \"xGF'F)" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "de := p*q*diff(y(x),x$2)+(p*r+q*s)* diff(y(x),x)+r*s*y(x)=exp(-r/q*x);\ndesolveCC(de,y(x),info=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(*(%\"pG\"\"\"%\"qGF)-%%diffG6 $-%\"yG6#%\"xG-%\"$G6$F1\"\"#F)F)*&,&*&F(F)%\"rGF)F)*&F*F)%\"sGF)F)F)- F,6$F.F1F)F)*(F9F)F;F)F.F)F)-%$expG6#,$*(F9F)F*!\"\"F1F)FD" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,(*(%\"pG\"\"\"% \"qGF()%\"mG\"\"#F(F(*&,&*&F'F(%\"rGF(F(*&F)F(%\"sGF(F(F(F+F(F(*&F0F(F 2F(F(\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G,$*&%\"pG!\"\"%\"sG\"\"\"F',$*&%\"qGF'%\"r GF)F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$% " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&-%$expG6#,$*(%\"rG\" \"\"%\"qG!\"\"F'F0F2F0%$_C2GF0F0*&-F+6#,$*(%\"pGF2%\"sGF0F'F0F2F0%$_C1 GF0F0*(F*F0F'F0,&*&F9F0F/F0F0*&F1F0F:F0F2F2F2" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 8" }}{PARA 257 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2*a;" "6#,&*(%\"dG\"\"#%\"yG\" \"\"*&F%F(*$%\"xGF&F(!\"\"F(*&F&F(%\"aGF(F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+[a^2+b^2]*y = x;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F' *&7#,&*$%\"aG\"\"#F'*$%\"bGF/F'F'%\"yGF'F'%\"xG" }{TEXT -1 8 ", where \+ " }{TEXT 268 1 "a" }{TEXT -1 5 " and " }{TEXT 269 1 "b" }{TEXT -1 14 " are positive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "assume(a>0):assume(b>0);\nde := diff(y(x),x$2)+ 2*a*diff(y(x),x)+(a^2+b^2)*y(x)=x;\ndesolveCC(de,info=true);\na := 'a' : b := 'b':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\" yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2%#a|irGF2-F(6$F*F-F2F2*&,&*$)F4F1F 2F2*$)%#b|irGF1F2F2F2F*F2F2F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8aux iliary~equation~.~.~G/,**$)%\"mG\"\"#\"\"\"F**(F)F*%#a|irGF*F(F*F**$)F ,F)F*F**$)%#b|irGF)F*F*\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G,&%#a|irG!\"\"*&%#b|irG \"\"\"^#F)F)F),&F%F&*&^#F&F)F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# %!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%6#F/F,F@F,-%$cosGFGF,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG, (*(-%$expG6#,$*&%#a|irG\"\"\"F'F0!\"\"F0-%$sinG6#*&%#b|irGF0F'F0F0%$_C 2GF0F0*(F*F0-%$cosGF4F0%$_C1GF0F0*&,(*&F'F0)F/\"\"#F0F0*&F'F0)F6F@F0F0 *&F@F0F/F0F1F0,&*$F?F0F0*$FBF0F0!\"#F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 9" }}{PARA 257 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "d^2*y/(d*x^2)+2*a;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(* $%\"xGF&F(!\"\"F(*&F&F(%\"aGF(F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/d x+[a^2+b^2]*y = exp(-a*x)*cos(b*x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*& 7#,&*$%\"aG\"\"#F'*$%\"bGF/F'F'%\"yGF'F'*&-%$expG6#,$*&F.F'%\"xGF'F)F' -%$cosG6#*&F1F'F9F'F'" }{TEXT -1 8 ", where " }{TEXT 270 1 "a" }{TEXT -1 5 " and " }{TEXT 271 1 "b" }{TEXT -1 14 " are positive." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "ass ume(a>0):assume(b>0);\nde := diff(y(x),x$2)+2*a*diff(y(x),x)+(a^2+b^2) *y(x)=exp(-a*x)*cos(b*x);\ndesolveCC(de,info=true);\na := 'a': b := 'b ':" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG -%\"$G6$F-\"\"#\"\"\"*(F1F2%#a|irGF2-F(6$F*F-F2F2*&,&*$)F4F1F2F2*$)%#b |irGF1F2F2F2F*F2F2*&-%$expG6#,$*&F4F2F-F2!\"\"F2-%$cosG6#*&F=F2F-F2F2 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%8auxiliary~equation~.~.~G/,**$)% \"mG\"\"#\"\"\"F**(F)F*%#a|irGF*F(F*F**$)F,F)F*F**$)%#b|irGF)F*F*\"\"! " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%%+roots~.~.~G,&%#a|irG!\"\"*&%#b|irG\"\"\"^#F)F)F),&F%F&*&^#F&F) F(F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%F 1F,-F56$,$*&#F,F:F,*&F3F.-F0F?F,F,F.F-F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%)~~~~~~~=G,$*&#\"\"\"\"\"%F'*(-%$expG6#,$*&%#a|irGF'% \"xGF'!\"\"F',&-%$cosG6#*&%#b|irGF'F0F'F'**\"\"#F'-%$sinGF5F'F7F'F0F'F 'F'F7!\"#F'F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,$*&#\"\"\"\"\"%F&*(%#b|irG!\"#-%$expG6#,$*&%#a|irGF &%\"xGF&!\"\"F&-%$cosG6#*&F)F&F1F&F&F&F&%fo~in~the~particular~solution ~can~be~absorbed~into~the~complementary~solutionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,( *&#\"\"\"\"\"#F+**%#b|irG!\"\"-%$expG6#,$*&%#a|irGF+F'F+F/F+-%$sinG6#* &F.F+F'F+F+F'F+F+F+*(&%\"CG6#F+F+F0F+F6F+F+*(&F<6#F,F+F0F+-%$cosGF8F+F +" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"y G6#%\"xG,(*(-%$expG6#,$*&%#a|irG\"\"\"F'F0!\"\"F0-%$sinG6#*&%#b|irGF0F 'F0F0%$_C2GF0F0*(F*F0-%$cosGF4F0%$_C1GF0F0*&#F0\"\"#F0*(F*F0,&F9F0*(F2 F0F6F0F'F0F0F0F6!\"#F0F0" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 67 "Find the general solutions of th e following differential equations." }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 259 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2 *y/(d*x^2)-2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F ," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-8*y = 2*x^2-7*x-2" "6#/,&*&%# dyG\"\"\"%#dxG!\"\"F'*&\"\")F'%\"yGF'F),(*&\"\"#F'*$%\"xGF/F'F'*&\"\"( F'F1F'F)F/F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 40 "_________ ______________________________ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 40 "_______________________________________ " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 2 "Q2" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+dy/dx-6*y = 2*x;" "6#/,(*(%\"dG\"\"#%\"yG\"\"\"*&F &F)*$%\"xGF'F)!\"\"F)*&%#dyGF)%#dxGF-F)*&\"\"'F)F(F)F-*&F'F)F,F)" } {TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________ ________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 2 "Q3" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{XPPEDIT 18 0 "d ^2*y/(d*x^2)-5;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F( \"\"&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx = 2*x^3-4*x^2-x+6;" "6# /*&%#dyG\"\"\"%#dxG!\"\",**&\"\"#F&*$%\"xG\"\"$F&F&*&\"\"%F&*$F-F+F&F( F-F(\"\"'F&" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "_________ ______________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________ ________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q4" }}{PARA 0 "" 0 "" {TEXT -1 5 " " } {XPPEDIT 18 0 "4;" "6#\"\"%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2*y/(d* x^2)-4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"%F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx-3*y = cos(2*x);" "6#/,&*&%#dyG\" \"\"%#dxG!\"\"F'*&\"\"$F'%\"yGF'F)-%$cosG6#*&\"\"#F'%\"xGF'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "_____________________________ __________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q5" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "d^ 2*y/(d*x^2)+4;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\" \"%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+4*y = 3*exp(-2*x);" "6#/, &*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"%F'%\"yGF'F'*&\"\"$F'-%$expG6#,$*&\"\" #F'%\"xGF'F)F'" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "______ _________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________ ________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q6" }}{PARA 0 "" 0 "" {TEXT -1 6 " 2 " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$ %\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 2 " " }{XPPEDIT 18 0 "dy/dx-2*y = \+ 14*x^2-4*x-11;" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F),(*& \"#9F'*$%\"xGF+F'F'*&\"\"%F'F1F'F)\"#6F)" }{TEXT -1 2 " " }}{PARA 0 " " 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "__ _____________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q7" }} {PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(\"\"$F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = sin(exp(x));" "6#/,&*&%#dyG\"\"\" %#dxG!\"\"F'*&\"\"#F'%\"yGF'F'-%$sinG6#-%$expG6#%\"xG" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 39 "_____________________________________ __" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 " " {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q8" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 " d^2*y/(d*x^2)-3;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F( \"\"$F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = exp(3*x)/(1+exp(x ));" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\"\"#F'%\"yGF'F'*&-%$expG6#*&\" \"$F'%\"xGF'F',&F'F'-F/6#F3F'F)" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "____________ ___________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 2 "Q9" }}{PARA 0 "" 0 "" {TEXT -1 7 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-2;" "6#,&*(%\"dG\" \"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+2*y = exp(x)*sec(x);" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&\" \"#F'%\"yGF'F'*&-%$expG6#%\"xGF'-%$secG6#F1F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q10" } }{PARA 0 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2; " "6#,&*(%\"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+y = exp(-x)*ln(x);" "6#/,&*&%#dyG\"\"\"%#d xG!\"\"F'%\"yGF'*&-%$expG6#,$%\"xGF)F'-%#lnG6#F0F'" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 39 "_______________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 256 "" 0 "" {TEXT 267 14 "Test exa mples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 54 "The following examples were used for testing purposes. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 9 "desolveCC" }{TEXT -1 46 ": examples involving 'awkward' real constant s " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ":" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "de := diff(y(x),x$2)-5*diff( y(x),x)+Pi*y(x)=8*exp(x);\ndesolveCC(de,info=false);\ndsolve(de,y(x)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\"xG-% \"$G6$F-\"\"#\"\"\"*&\"\"&F2-F(6$F*F-F2!\"\"*&%#PiGF2F*F2F2,$-%$expGF, \"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&-%$expGF&\" \"\",&!\"%F,%#PiGF,!\"\"\"\")*&&%\"CG6#F,F,-F+6#*&,&#\"\"&\"\"#F,*&#F, F " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "de := diff(y (x),x$2)-2*diff(y(x),x)+Pi*y(x)=cos(x);\ndesolveCC(de,info=false);\nds olve(de,y(x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$- %\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"*&%#PiGF2F*F2F2 -%$cosGF," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&,&-%$si nGF&\"\"#-%$cosGF&\"\"\"F0,(\"\"&F0*&F-F0%#PiGF0!\"\"*$)F4F-F0F0F5F5*( &%\"CG6#F0F0-%$expGF&F0-F,6#*&-%%sqrtG6#,&F4F0F0F5F0F'F0F0F0*(&F:6#F-F 0F " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "de := diff(y(x),x$2)-2*sin(2)*diff (y(x),x)+sin(2)^2*y(x)=3*x;\ndesolveCC(de);\nevalf(%);\ndsolve(de,y(x) );\nevalf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-% \"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*(F1F2-%$sinG6#F1F2-F(6$F*F-F2!\"\"*&) F4F1F2F*F2F2,$F-\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"x G,(*&,&*&-%$sinG6#\"\"#\"\"\"F'F0!#7\"#C!\"\"F0,&F,!\"$-F-6#\"\"'F0F3F 0*&&%\"CG6#F0F0-%$expG6#F+F0F0*(&F;F.F0F'F0F=F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,*F'$\"+68NGO!\"*$\"+Y*f0)zF+\"\"\"*&&%\" CG6#F.F.-%$expG6#,$F'$\"+oU(H4*!#5F.F.*(&F16#\"\"#F.F'F.F3F.F." }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,(*&-%$expG6#*&-%$sinG6# \"\"#\"\"\"F'F2F2%$_C2GF2F2*(F*F2F'F2%$_C1GF2F2*(,&!\"'F2*(\"\"$F2F.F2 F'F2!\"\"F2,&F;F2*$)-%$cosGF0F1F2F2F;F.F;F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,**&-%$expG6#,$F'$\"+oU(H4*!#5\"\"\"%$_C2 GF1F1*(F*F1F'F1%$_C1GF1F1$\"+W*f0)z!\"*F1*&$\"+58NGOF7F1F'F1F1" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "If a and b are real \+ numbers with " }{XPPEDIT 18 0 "b>a" "6#2%\"aG%\"bG" }{TEXT -1 59 ", th en the auxiliary equation of the differential equation " }}{PARA 257 " " 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(% \"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+(1+b-a)*y = 0" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&, (F'F'%\"bGF'%\"aGF)F'%\"yGF'F'\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 18 "has complex roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 121 "a := 'a': b := 'b':\nassume (a%#deG /,(-%%diffG6$-%\"yG6#%\"xG-%\"$G6$F-\"\"#\"\"\"*&F1F2-F(6$F*F-F2!\"\"* &,(F2F2%#b|irGF2%#a|irGF6F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(&%\"CG6#\"\"\"F--%$expGF&F--%$sinG6#,$*&-%%sqr tG6#,&%#b|irGF-%#a|irG!\"\"F-F'F-\"\"#F-F-*(&F+6#F " 0 "" {MPLTEXT 1 0 162 "a:=sin(Pi/3):\nb:=sin(314159265358979323846264338327 9502884197169399375105820974944592307816406286208998628034825342117068 /(3*10^99)):\nevalf(b-a,110);\nsignum(b-a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"*\"*=`(H!$5\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" \"\"" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 " The auxiliary equation of the differential equation " }}{PARA 257 "" 0 "" {TEXT -1 6 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"d G\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " } {XPPEDIT 18 0 "dy/dx+(1+b-a)*y = 0" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&, (F'F'%\"bGF'%\"aGF)F'%\"yGF'F'\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 " " {TEXT -1 24 "again has complex roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "de := diff(y(x),x$2)-2* diff(y(x),x)+(1+b-a)*y(x)=0:\ndesolveCC(de,y(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG,&*(&%\"CG6#\"\"\"F--%$expGF&F--%$sinG6#, $*&-%%sqrtG6#,&-F16##\"^qn#HbL1(3ql\\Abr:5ap2[htV_XwP%)\\BH\\5sv)>e%3m :'4$[uRj\")R&y\"^q++++++++++++++++++++++++++++++++++++++++++++++++](\" \"%*&\"\"#F--F66#\"\"$F-!\"\"F-F'F-#F-F@F-F-*(&F+6#F@F-F.F--%$cosGF2F- F-" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/ -%\"yG6#%\"xG,&*(%$_C1G\"\"\"-%$expGF&F+-%$sinG6#,$*&-%%sqrtG6#,&-F/6# #\"^qn#HbL1(3ql\\Abr:5ap2[htV_XwP%)\\BH\\5sv)>e%3m:'4$[uRj\")R&y\"^q++ ++++++++++++++++++++++++++++++++++++++++++++++](\"\"%*&\"\"#F+-F46#\" \"$F+!\"\"F+F'F+#F+F>F+F+*(%$_C2GF+F,F+-%$cosGF0F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Define two nearby numb ers a and b such that " }{XPPEDIT 18 0 "a " 0 " " {MPLTEXT 1 0 174 "a:=BesselJ(0,Pi/3):\nb:=BesselJ(0,3141592653589793 2384626433832795028841971693993751058209749445923078164062862089986280 34825342117068/(3*10^99)):\nevalf(b-a,110);\nsignum(b-a);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "The auxiliary equ ation of the differential equation " }}{PARA 257 "" 0 "" {TEXT -1 6 " \+ " }{XPPEDIT 18 0 "d^2*y/(d*x^2)+2;" "6#,&*(%\"dG\"\"#%\"yG\"\"\"* &F%F(*$%\"xGF&F(!\"\"F(F&F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+(1+ b-a)*y = 0" "6#/,&*&%#dyG\"\"\"%#dxG!\"\"F'*&,(F'F'%\"bGF'%\"aGF)F'%\" yGF'F'\"\"!" }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "again has complex roots." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 71 "de := diff(y(x),x$2)-2*diff(y(x),x)+(1+b-a)*y( x)=0:\ndesolveCC(de,y(x));" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "dsolve(de,y(x));" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 5" }}{PARA 257 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "d^2*y/(d*x^2)-``(C[1]+C[3]);" "6#,&*(% \"dG\"\"#%\"yG\"\"\"*&F%F(*$%\"xGF&F(!\"\"F(-%!G6#,&&%\"CG6#F(F(&F26# \"\"$F(F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "dy/dx+C[1]*C[3]*y = 0;" "6# /,&*&%#dyG\"\"\"%#dxG!\"\"F'*(&%\"CG6#F'F'&F,6#\"\"$F'%\"yGF'F'\"\"!" }{TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "de := diff (y(x),x$2)-(C[1]+C[3])*diff(y(x),x)+C[1]*C[3]*y(x)=0;\ndesolveCC(de,y( x));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#deG/,(-%%diffG6$-%\"yG6#%\" xG-%\"$G6$F-\"\"#\"\"\"*&,&&%\"CG6#F2F2&F66#\"\"$F2F2-F(6$F*F-F2!\"\"* (F5F2F8F2F*F2F2\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"yG6#%\"xG ,&*&&%\"CG6#\"\"#\"\"\"-%$expG6#*&&F+6#F.F.F'F.F.F.*&&F+6#\"\"%F.-F06# *&&F+6#\"\"$F.F'F.F.F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }